2000 USAMO Problems/Problem 1
Call a real-valued function very convex if
holds for all real numbers and . Prove that no very convex function exists.
Let , and substitute . Then a function is very convex if , or rearranging,
Let , which is the slope of the secant between . Let be arbitrarily small; then it follows that , . Summing these inequalities yields . As (but , so is still arbitrarily small), we have . This implies that in the vicinity of any , the function becomes vertical, which contradicts the definition of a function. Hence no very convex function exists.
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